>>TBH, at this point I don't see how you can progress this without more information as to what is actually happening and, AFAICS, that means either logging as above or monitoring running processes and/or file access using something like Sysinternals.....
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>Can we assume that this cannot be related to a query in progress with SQL Server? Because, everyone is within the same situation. Web sites process hits against a database. In our case, we are targeting at SQL Server. So, if others do not have this situation, then, it has to be related to something else.

I don't know how you are managing connections or what SQL operations takes place at startup (or shutdown-see below) but, normally, I wouldn't think SQL should be a problem...

>Also, when it happened once today, I killed the wp3p.exe process and it went ok. The Web site started to respond immediately. I mean, right in the next fraction of a second that is. So, this means the new DLL is loaded in memory. But, the Web site cannot deliver a respond to the request. In the last two times this has happening, I have been able to make it go by killing the wp3p.exe process.

Given that the new instance kicks in so quickly then that sounds as if the process that you are killing is the 'old' instance and that it is not shutting down properly. But that, in itself, shouldn't prevent the new instance from running since, by definition, it has to take over the acceptance of new requests to allow the old instance to become idle. Maybe some sort of deadly embrace : the old one can't shut down because the new one is locking some resource and the new one can't complete initializing because the old one is doing the same? Again all just supposition (and getting a bit esoteric).

I'd really suggest putting logging code into the Application startup and shutdown events so that you can get a clearer picture of what is happening ( maybe give the instance a Guid on startup so that you can differentiate between the two instances)